Parametrization

1. Velocity Vector: \[ \underline{r}'(t)=\underline{r}' = \begin{pmatrix} x'(t)\\y'(t) \end{pmatrix} \]
2. Accelleration Vector: \[ \underline{r}''(t)=\underline{r}'' = \begin{pmatrix} x''(t)\\y''(t) \end{pmatrix} \]
3. Scalar Velocity: \[ v(t) = \sqrt{x'^2+y'^2} = \left(\underline{r}' \cdot \underline{r}'\right)^{1/2} \]
4. Determinant: \[ det(t) = \begin{vmatrix} x' & x''\\ y' & y'' \end{vmatrix} = \underline{r}'' \cdot \underline{\widehat r}' \] Projection of $\underline{r}''$ on $\underline{\widehat r}'$.
5. Regular Points: $\underline{r}'(t) \neq \underline{0}$.
6. Frenet System, $v(t)>0$. Origin in $\underline{r}(t)$ and Orthonormal Vectors: \[ \underline{t}(t) = \frac{\underline{r}'}{v}, \qquad \underline{n}(t) = \frac{\underline{\widehat r}'(t)}{v} = \underline{\widehat t} \]
7. Natural Angle, $\theta(t)$: \[ \underline{t}(t) = \begin{pmatrix} \cos{\theta(t)} \\ \sin{\theta(t)} \end{pmatrix} \]
8. Derivatives, Frenet Equations: \[ \underline{t}'(t) = \theta'(t) \underline{n}, \qquad \underline{n}'(t) = -\theta'(t) \underline{t} \]
9. Angular Velocity: \[ \omega(t) = \theta'(t) = \underline{t}' \cdot \underline{n} = \frac { \underline{r}'' \cdot \underline{\widehat r}' }{v(t)^2} ~\left[ s^{-1}\right] \]
10. Curvature: \[ \kappa(t) = \frac { \underline{r}'' \cdot \underline{\widehat r}' }{v(t)^3} ~\left[ m^{-1}\right] \]
11. Curvature Ratio: \[ \rho(t) = \frac{1}{\kappa(t)} = \frac{v(t)^3} { \underline{r}'' \cdot \underline{\widehat r}' } ~\left[ m \right] \]
12. Center of Curvature, Evolute: \[ \underline{c}(t) = \underline{r}(t) + \rho(t) \underline{n}(t) = \underline{r}(t) + \omega(t) \underline{\widehat r}'(t) \]
13. Oscullating Circle: \[ \left(\underline{r}(t), \rho(t)\right) \] Best Second Order Approximation.
14. Taylor Formula: \[ \underline{r}(t) = \underline{r}(t_0) +(t-t_0) \underline{r}'(t_0) +\frac{1}{2}(t-t_0)^2 \underline{r}''(t_0) + \underline{\varepsilon}(t) \]
15. \[ c=\underline{e}(t) \cdot \underline{e}(t) \Rightarrow 0=2\underline{e}'(t) \cdot \underline{e}(t) \Leftrightarrow \underline{e}'(t) \perp \underline{e}(t) \]